Download Polarimetric imaging of crystals - Werner Kaminsky

Polarimetric imaging of crystals
Werner Kaminsky, Kacey Claborn and Bart Kahr
Department of Chemistry, Box 351700, University of Washington, Seattle, Washington,
USA 98195-1700
Received 5th February 2002
First published as an Advance Article on the web 28th September 2004
Classical crystal optics has recently undergone a renaissance as developments in optical microscopy and
polarimetry, enabled in part by sensitive imaging CCD cameras and personal computers, now permit the
analytical separation of various optical effects that are otherwise convolved in polarized light micrographs. In
this tutorial review, we review recent developments in the measurement of the principal crystallo-optical
quantities including linear birefringence, linear dichroism, circular birefringence, and circular dichroism, as well
as new effects in crystal optics encountered in unusual mixed crystals. The new microscopies and polarimetries
are applied to problems of crystallographic twinning, phase transformations, stress birefringence, symmetry
reduction, and the design of new crystalline materials.
1. Introduction
DOI: 10.1039/b201314m
Classical crystal optics is the science that involves the
simultaneous reckoning of four phenomena, linear birefringence (LB), linear dichroism (LD), circular birefringence (CB),
and circular dichroism (CD).1 These are, respectively, the
anisotropy in refraction, the anisotropy in absorption, the
difference in the refractive index of left and right circularly
514
polarized light, and the difference in absorption of left and
right circularly polarized light. Generally speaking, when more
than one of these effects are present at the same time, they
are convolved in the elliptical polarization state of the light
propagating through the medium. Consequently, complex
materials between polarizing elements often present a dazzling
array of interference colors whose value is mainly decorative.
Polarized light micrographs frequently win scientific art
Werner Kaminsky was born in Cologne,
Germany in 1959. He was educated at the
Institut für Kristallographie at the university in that city, receiving his diploma,
his PhD in crystal physics under the
supervision of Siegfried Haussühl, and
ultimately his habilitation in 2000.
Between 1994 and 1999 he spent several
periods as postdoctoral associate, tutor,
and lecturer at Oxford University working
with A. Michael Glazer on the development of new methods for the analysis of
the optical properties of crystals. For this
work he received the Philips physical
crystallography prize from the British
Association of Crystallography. In 2000
he joined the faculty of the Department
of Chemistry at the University of
Washington, Seattle where he is currently
Research Associate Professor.
Bart Kahr was born in New York City in
1961. He attended Middlebury College in
Vermont where he was introduced to
research in chemistry by I. David
Reingold. His graduate studies of the
stereochemistry of unusual molecules
with Kurt Mislow at Princeton University
were followed in 1988 by postdoctoral
research in crystal chemistry at Yale
University in the laboratory of J. Michael
McBride. In 1990 he joined the faculty of
Purdue University and was named a US
National Science Foundation Young
Investigator. He moved to the University
of Washington, Seattle, in 1997 where he
is currently Professor of Chemistry. His
research group studies the growth, structure, and physical properties of crystalline
materials, as well as the history of
crystallography.
Werner Kaminsky
Kacey Claborn
Bart Kahr
Kacey Claborn was born in Lindsay,
California in 1974. She received her BS
in crystallography at the University of
Washington in 2001 and expects to
complete her PhD at the same institution
in 2005. Her research with Bart Kahr and
Werner Kaminsky is focused on measurements of the chiroptical properties of
crystalline materials. She is the recipient
of a graduate fellowship from the organic
chemistry division of the American Chemical Society.
Chem. Soc. Rev., 2004, 33, 514–525
This journal is ß The Royal Society of Chemistry 2004
contests and grace the covers of chemical company catalogs but
often mask the underlying nature of the illuminated material.
The analysis of crystals with visible light was a highly
developed art prior to the discovery of X-ray diffraction in
1912. At that time, crystallographers shifted their attention in
large measure to the determination of the positions of atoms
in crystals. Today, X-ray crystallography is a highly developed
art. Even the analyses of many biopolymers have been reduced
to routine practice. Concurrently, developments in polarimetry, polarized light microscopy, and the advent of CCD
cameras coupled to micro-computers have signaled a renaissance in the unfinished science of crystal optics. Only within the
past 20 years have the tools been developed for separating the
various optical effects from one another thereby maximizing
the information about crystal structure and growth that is in
principle contained in visible light micrographs. Here, we
review these developments with illustrations from our recent
work. We have focused these illustrations on a small number of
substances – alkali sulfates, sodium halates, and 1,8-dihydroxyanthraquinone – so as to draw various methodologies together
in visits to recurrent materials.
was proposed in 1982 by Maestre and Katz who adapted a
Carey spectropolarimeter to a microscope9 for single point
measurements of the CD spectra of chromatin. They faced
instrumental artifacts10 arising from electronic polarization
modulators in commercial instruments that typically generate
sinusoidally varying polarization states,11 thereby introducing
a small admixture of linearly polarized light into the circularly
polarized output. Residual ellipticity, when coupled with the
LB and LD of ordered media, generates artifactual CD
signals.12,13 Strain in photoelastic modulators (PEMs) compounds these artifacts.14 Attempts have been made to skirt
these problems by adding additional modulators,15 rotating the
sample,16 performing complex analytical transformations of
independent chiroptical measurements,17 or choosing components that minimize polarization biases.18 Nevertheless, a
reliable device for imaging CD has eluded investigators for the
most part. Recently, CD images of d-camphorsulfonic acid
films with a spatial resolution of v 1 m were obtained by
Yamada et al.19 They employed a polarizing undulator applied
to near UV synchrotron radiation.
2. Differential polarization imaging
3. Tools
The work described herein has so many antecedents as to
require a review of nearly two centuries of scholarly work
devoted to polarimetry and optical crystallography. Some of
this is contained within several review articles2,40,44 but will not
be addressed here in order to ground this work in recent
history. The development of differential polarization imaging
by Maestre, Tinoco, Bustamante and coworkers in the 1980s
serves as an adequate introduction to the tools that we will
introduce in the next section. As a foil, coming just before the
advent of CCD detectors, it sets off the work that succeeded it.
All of the new methods that we have put to use in the study of
crystals involve the analysis of intensity signals as a function of
the modulation of input polarization. Applications of such a
strategy in conjunction with microscopy to form images of
LD and CD (or related quantities) can be credited to the
aforementioned group.
The relevant body of research was begun in 1985 when
Mickols et al. introduced the imaging differential polarization
microscope.3 In this device various voltages were applied to a
KH2PO4 crystal in a Pockels cell in order to modulate between
orthogonal polarization states, or between left and right
circular polarization states at a frequency of less than 1 kHz.
The input polarization was then passed through the sample on
a microscope stage. The transmission was analyzed by a linear
256 diode array detector that was translated to produce a 2D
image. The difference between the orthogonal polarization
states, and the left and right circular polarization states, could
then be plotted, in principle, as LD and CD, respectively. This
device was not employed in the analysis of single crystals, but
rather in the analysis of polycrystalline biological structures
such as intracellular polymeric hemoglobin aligned in subjects
burdened by sickle cell anemia.4 LD and circular differential
images5 were produced also for spermatocyte nuclei. Here, a
sample micro-positioning stage was used to make images in
conjunction with a photo multiplier tube detector.6 Bustamante
and coworkers introduced an image dissector to scan the image
of a sample on a fixed stage, focusing especially on sickled
hemoglobin.7
By 1995, the use of a linear 1024 diode array was responsible
for much greater detail in the LD images, thereby permitting
the sub-classification of sickled hemoglobin cells.8 CD imaging
microscopy did not develop in tandem with LD imaging
microscopy, the former having been comparatively troublesome. The idea of a CD microscope for anisotropic samples
3.1. Jones calculus
A variety of methods have been developed for tracking the
polarization state of light as it passes through successive optical
elements and complex samples. These are the Poincaré sphere,20
the Mueller calculus,21 and the Jones calculus.22 These tools, and
their relative advantages, have been treated in detail elsewhere.23
Here, we give an overview of the Jones formalism, the only
method that we will make use of in the following.
The x and y components of an electric vibration can be
represented by a two-element column vector called the Jones
vector after its inventor R. Clark Jones. If the amplitudes of the
x and y components are given as Axeiwx and Ayeiwy then the
Jones vectors (J) for linear polarized light along x (Jx-pol),
elliptically polarized light (Jellipt), as well as right (Jrcp) and left
(Jlcp) circularly polarized light are:
"
#
#
Ax eiwx
Ax eiwx
Jellipt ~
Jx{pol ~
Ay eiwy
0
"
#
"
#
Aeiw
Aeiw
Jlcp ~
:
Jrcp ~
Aei(wzp=2)
Aei(w{p=2)
"
The Jones vectors may be normalized such that J**J ~
Ax2 1 Ay2 ~ 1. The resulting expressions are invariant to a
constant phase added to both components. Thus, simpler
equivalent vectors follow:
Ax
1
1
Jx{pol ~
Jellipt ~ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
iwy
0
A2x zA2y Ay e
1
1 1
1
Jrcp ~ pffiffiffi
Jlcp ~ pffiffiffi
:
2 i
2 {i
Given an input polarization state, Jin, the output polarization state, Jout, can be reckoned with a square matrix (M) that
represents the sample, or some optical element such that:
Jout ~ M Jin
The matrices corresponding to a linear polarizer along x, an
arbitrary retardation plate inducing a phase shift d ~ 2pLDn/l
(L is the sample thickness, Dn is the birefringence and l is the
wavelength) of the emergent light, and a linear polarizer
Chem. Soc. Rev., 2004, 33, 514–525
515
oriented at an arbitrary angle h are:
"
#
"
#
1 0
0
eid=2
Mretard ~
Mx{pol ~
0 0
0
e{id=2
"
#
cos2 h
sin h cos h
T
,
Mh{pol ~Rh Mx{pol Rh ~
sin h cos h
sin2 h
"
#
cos h
sin h
Rh ~
{ sin h cos h
Below, the Jones formalism is used to construct right
circularly polarized light from a quarter wave plate acting on
linearly polarized light oriented at 45u with respect to the eigen
modes of the wave plate. A constant phase has been added to
simplify the expression.
"
#
1
1
pffiffiffi
Jout~
~
R45
2 {1
0
0
e{ip=4
0
e{ip=4
:
" #
"
#
"
#
eip=4
eip=4{ip=4
1
1
1 1
~Jrcp
~ pffiffiffi
: pffiffiffi
~ pffiffiffi
2 {e{ip=4
2 {e{ip=4{ip=4
2 i
"
eip=4
0
#
" #
1
"
eip=4
0
#
3.2. Metripol
In order to quantify linear anisotropies we have employed the
rotating polarizer technique as embodied in the Metripol
microscope.24 The optical train consists of a filter as
monochrometer, a mechanically rotating polarizer, a sample
inducing the phase shift d, a quarter wave retarder and a
polarizer aligned at 45u towards the quarter wave plate’s eigen
ray directions:
A’ ~ M45-polMl/4MsampleRhJx
The position of the sample with respect to the rotating
polarizer is defined by the angle h, h ~ a 2 w, where a is the
rotation angle of the polarizer and w the angle between the slow
vibration direction of the sample and the polarizer when a ~ 0.
The amplitude of the emergent light form A’ is then given as:
3
2 p
32 d
"
#
"
#
i
i
6
cos h
sin h
e4
0 7
0 7
e2
1 1 1 .6
7
6
0
6
7
.
.
A~
7.
p56
4
2 1 1 4
d 5 { sin h cos h
{i
{i
0 e 4
0 e 2
" # "
#
ix
{ix
1
1 cos he { sin he
~
d p
2 { sin he{ix z cos heix
0
x~( z )
2 4
From the amplitudes, the normalized intensity I/I 0 is found
directly as:
I
1
~A0 .A0 ~2. cos h eix { sin h e{ix cos h e{ix { sin h eix
I0
4
1 2
~ cos h{ cos h sin h e2ix ze{2ix z sin2 h
2
1
~ ½1{ cos h sin h 2 cos (2x)
2
1
~ ½1z sin 2h sin d
2
1
~ ½1z sin 2(a{w) sin d
2
Fig. 1 shows how the intensity varies as a function of the
polarizer angle and the parameters d and w.
516
Chem. Soc. Rev., 2004, 33, 514–525
Fig. 1 Dependence of signal I/I 0 on the rotation of the polarizer angle
(a). The amplitude is related to the phase difference (d), the phase shift
gives the optical orientation (w), and the offset gives the transmittance.
In earlier incarnations of the rotating polarizer technique,25
the phase velocity was taken to be a ~ vt. This latter
description, however, was incompatible with CCD imaging in
which each pixel must be computed independently. In the
Metripol method, intensity measurements at discrete steps (ai)
generate expressions that are easily converted to linear
polynomials. Data collected over full periods yield Fourier
coefficients from which the variable parameters are extracted
analytically without any computationally intensive matrix
inversions.
Simple modifications of the optical train permit the measurement of LD and of OR in special cases. The anisotropic
absorption may be experimentally accessed by removing the
quarter wave plate and analyzer. The resulting optical train is
represented by:
"
#
0
tx eid=2
A0 ~Msample Rh Jx , Msample ~
t
0
ty e{id=2
1
t~e4(ax zay )
e
tx=y ~e+2
(ttx)2 and (tty)2 are the transmission coefficients of the sample
along the eigen ray directions. e ~ (ax 2 ay)/2 where ax/y are the
absorption coefficients. Applying the same procedure as
before, neglecting overall absorption, t2, we find an expression
for the intensity that is independent of d:
I
~t2x cos2 hzt2y sin2 h,
I0
which may be recast in the more useful form:
1
I
1
~ t2x zt2y z t2x {t2y cosð2hÞ:
I0 2
2
I
~ cosh ez cosð2hÞ sinh e,
I0
or
I=I0
~1z cosð2hÞ tanh e:
cosh e
In a measurement, the detected intensities, I, are normalized
to the average intensities, I’ ~ I/vIw.
I0
:½1z cosð2hÞtanhe:
I0
The amplitude in Fig. 1 is then determined by tanh e, a
measure of the scaled differential transmission, or LD.
A second variation in the optical train can be used to probe
optically active materials in the absence of birefringence,
requiring only the removal of the quarter wave plate:
0
A0 ~Mpol{y Msample Rh Jx ~
:
{ sin h
In this case, Msample is the unit matrix and the only
non-vanishing component of Mpol-y is M22 ~ 1, giving the
intensity expression:
I
1
~ sin2 h~ (1{ cos 2h):
I0
2
3.3. HAUP and S-HAUP
To measure OR (circular birefringence, DnCB ~ nR 2 nL) along
a general direction of a crystal one must deconvolve the
influence of LB (DnLB ~ n@ 2 n’). Although the intensity of
light passing through a polarizer, chiral anisotropic sample,
and analyzer contains the necessary information for extracting
OR, implementation of this idea prior to the invention of
electrophotometry and stable, high-intensity light sources was
impossible. For generations, most researchers conceded that
measuring OR in the presence of large LB was not a realistic
goal.
In 1983, Uesu and Kobayashi pushed through this impasse
by using lasers, photon counting, and computerized modulation of polarizer and analyzer orientations to determine OR in
crystals for directions off the optic axes.26 They called this
experiment HAUP (high accuracy universal polarimetry)
whose basic geometry is given in Fig. 2. Since we apply the
with
cos V { sin V
cos Y
RV ~
, RY ~
sin V
cos V
sin Y
cos h0 { sin h0
:
Rh ~
sin h0
cos h0
{ sin Y
,
cos Y
The result of these operations is written approximately as a
bi-quadratic polynomial that is normalized to the amplitudes
of Y 2 and V2:
I
~A .A~a0 za1 Vza2 Y za3 VY zV2 zY 2 ,
I0
with
1
g
Q
a1 ~{h(d)zQ(d)~(h0 { )( cos d{1)z 0 sin d,
d
2
d
1
g
Q
a2 ~h(d)zQ(d)~(h0 { )(1{ cos d)z 0 sin d,
2
d
d
1
a3 ~ cos d:
2
The first term (a0) is the overall offset in the intensity
measurement. Parameters Q and h are found from combinations of the parameters ai: Q 6 ((sin d)/d) ~ (a1 1 a2)/4;
h(h0, g) 6 ((cos d) 2 1) ~ (a2 2 a1)/4.
3.4. Circular extinction imaging
Fig. 2 The HAUP optical train. (1) light source, (2) polarizer, rotation
angle Y, (3) sample and translation stage with translation directions
t and x, (3a) extinction angle h of the birefringent cross-section
with refractive indices n’ and n@, (4) analyzer, rotation angle V, (5)
detector.
HAUP technique to heterogeneous crystals, the polarimeter
was fitted with a translation stage in order to produce
topographs of the optical parameters. In this way, maps
were made comprising 100 6 100 pixels at a resolution of
about 30 mm per pixel. In the Fourier analysis of the intensity
data, the apparent extinction angle convolved with circular
dichroism (h (h0, g)), the phase factor (d), and the apparent OR
(Q) were unfolded. We call the imaging experiment scanningHAUP (S-HAUP).
According to the Jones-matrix formalism,27 a sample that
shows LB (d ~ 2pDnLBL/l) and OR (Q), as well as CD (g) is
given by the following:
3
2
sin (d=2)
id=2
{2(Qzig)
e
7
6
d
7:
M(d,Q,g)~6
5
4
sin (d=2)
{id=2
2(Qzig)
e
d
In a HAUP experiment, the sample at extinction angle h0 is
placed between two orthogonal polarizers, which are rotated
about small angles Y and V (Fig. 2). The optical train is
represented by a string of matrices yielding the light amplitude
A from the rotation matrices for the polarizer (RY), analyzer
(RV), and sample (Rh0):
1
T 0 0
T
A~RV
RV Rh0 MRh0 RY
,
0 1
0
Reasonable sensitivity in CD measurements is commonly
achieved by rapid sampling concomitant with electronic light
modulation at rates of 50–100 kHz. Why then not add
contemporary CCD detection to make CD images? Operating
at less than 1 kHz, CCDs are incompatible with PEMs. While
others are trying to force compatibility28 by speeding up the
CCD29 or slowing down the modulation,30 these designs
remain constrained by limited spectral ranges, noise, and
parasitic ellipticities.31 On the other hand, physicists recently
built single point CD spectropolarimeters for anisotropic
media via schemes using mechanical light modulation with
photo multiplier tubes as detectors.32 Their devices were
extremely slow but nevertheless suited to large, homogeneous,
strong circularly dichroic crystals.
We built a circular dichroism imaging microscope (CDIM)
based on the apparently retrogressive mechanical modulation
of near perfect circularly polarized light (CPL) in conjunction
with CCD detection (Fig. 3). Signal to noise lost in slow
Fig. 3 Visible light circular dichroism imaging microscope (CDIM).
Schematic omits motors and mounts. (1) light source, (2) variable
interference filter, (3) depolarizer, (4) rotating polarizer, (5) tilting l/4compensator, (6) sample mount, (7) objective, (8) projector lens, (9)
depolarizer, (10) CCD-camera.
Chem. Soc. Rev., 2004, 33, 514–525
517
modulation (v 30 Hz) is regained by signal averaging with a
CCD camera. We abandoned the use of a broad band l/4 plate
in favor of a variable retarder that is adjusted so that it
functions as a perfect l/4 plate at each wavelength.
The Jones matrix for a birefringent and circular dichroic
sample is:12
2
3
sin x
ix
ig
e
6
x 7
7
M(d,g)~6
4
5
sin x
{ix
e
{ig
x
where x ~ d/2, and the CD is defined as g ~ 4(I1 2 I2)/I 0,
where I1 and I2 are right and left CPL, respectively. The
complex vectors A represent these light forms, where E0 is the
amplitude of the incoming light wave.
The intensity I’¡ is found from A¡’* ? A¡’, where A¡’ ~ M A¡.
The total intensity is then:
0
I+
sin x cos x
sin2 x
zg2 2
~1+2g
2
x
x
E0
The normalized intensity difference in a birefringent sample
then is found as:
IR0 {IL0
sin x cos x
sin d
~4g
~4g
:
I0
x
d
This expression describes the observed CD in a birefringent
sample provided that there is no parasitic linearly polarized
contribution to the incoming CPL.
4. Applications
4.1. Linear birefringence
Metripol was applied in its inaugural publication23 to the ferroelectric perovskite BaTiO3, synthetic diamond, and anorthosite.
In BaTiO3, tetragonal domains showed stress birefringence when
high-temperature cubic crystals were cooled to room temperature. Synthetic diamond showed stress birefringence associated
with defects and growth sector boundaries. One of these
celebrated images now graces the cover of the volume of the
International Tables for Crystallography devoted to physical
properties.33 The anorthosite micrographs dramatically
show the twin laws common to plagioclases. More recent
crystallographic applications of Metripol include illustrations
of the relaxor ferroelectric Na1/2Bi1/2TiO3,34 and of phase
transitions in Pb(Mg1/3Nb2/3)(12x)TixO335 and K2Mn2(SO4)3.36
Hollingsworth and Peterson used Metripol to demonstrate
domain switching in ferroelastic pseudo-hexagonal crystals of
2,10-undecanedione.37
4.1.1. K2SO4. A simple application and illustration of
Metripol is the screening of birefringent crystals. Shown in
Fig. 4 are three micrographs of the simple salt, K2SO4, grown
from aqueous solution on a glass slide. Fig. 4a shows the phase
difference between the two eigenmodes propagating through
the birefringent, orthorhombic crystals (space group Pmcn)
plotted as |sin d| in false color. |sin d| is naturally a periodic
function, but since K2SO4 is weakly birefringent and the
crystals are thin, the arc sine can be taken as the true value of
the phase factor. This micrograph is independent of crystal
orientation with respect to the coordinate system of the microscope. In an ordinary petrographic microscope, the intensity of
the transmitted light depends not only on the intrinsic
retardation but on the orientation of the individual crystallites.
In Fig. 4b, the orientations of cross sections of the optical
indicatrices are given as the angle of the slowest vibration
direction (the largest refractive index) measured counterclockwise from the horizontal axis. Here, one can plainly see
518
Chem. Soc. Rev., 2004, 33, 514–525
Fig. 4 Metripol micrographs of K2SO4 crystallites. (a) |sin d|, (b) w
(deg), (c) I/I 0 (%).
that the needles have distinct presentations. Some crystals are
lying on (010) whereas others are lying on (021). The two
‘‘yellow crystals’’ in Fig. 4b must have different presentations
as the [100] needles are pointing in orthogonal directions: the
only way they can both give the same false color is if the fast
and slow axes with respect to the microscope have been
inverted. It is easy to distinguish in the transmission map in
Fig. 4c those rectilinear crystals sitting on (010) from those with
oblique (021) presentations. The non-transmissive ‘‘purple’’
edges in Fig. 4c are merely inclined.
4.1.2. NaClxBr12xO3. Naturally, cubic crystals are optically
isotropic, but they can become birefringent when grown in the
presence of impurities that can reduce their symmetry38 either
by selectively occupying sites on growing surfaces that have
distinct presentations or by exerting stresses that can influence
the optical properties via the photoelastic, or piezo-optic
effects. In an ordinary polarizing microscope the birefringence
makes itself evident in complex patterns of interference
colors.39 Metripol analysis of slices can parse the various contributions to these ordinary micrographs.
A classic example of such so-called optically anomalous
crystals40 is a mixture of NaClO3 and NaBrO3. The cubic salts
are miscible in all proportions in the solid state. Mixed crystals
have cube {100} habits unless the mole fraction of BrO32 is
greater than 0.95 at which point the crystals become
tetrahedral. The {111} crystals can be prepared by modifying
the habit in the presence of S2O322(Fig. 5). Shown in Fig. 6 are
thin sections of mixed crystals of NaClxBr12xO3 that are
Fig. 5 Tetragonal and cubic habits of the sodium halates along with slices that indicate the bisected growth sectors. Dotted regions demarcate areas
represented in micrographs in Fig. 6. All of the crystal drawings herein were prepared by WinXMorph (W. Kaminsky, 2004).41
Fig. 6 Metripol LB micrographs of the sodium halates. (a,c) |sin d| maps. (b,d) Extinction maps. Orientation is measured counterclockwise from the
horizontal axis. (a,b) NaCl0.50Br0.50O3 grown in the presence of 5% Na2S2O3, 91 mm. (c,d) NaCl0.87Br0.13O3, 98 mm.
presented along a 3-fold crystallographic axis in {111} crystals
and the 2-fold crystallographic axis in {100} crystals (referred
to the idealized cubic system, space group P213). The
birefringence is evident in the |sin d| maps, Figs. 6a and 6c.
The corresponding extinction maps are shown in Figs. 6b and
6d. In the cubes, the central section represents a {100} sector
viewed along the growth direction. It has a much greater
birefringence than the surrounding regions that represent
sectors viewed normal to the growth directions. Despite these
large differences in the LB there is little difference in extinction.
In the tetrahedral crystals, the magnitude of |sin d| varies
slightly but chaotically across the crystal plate. Nevertheless,
the construction of the assemblage shows up brilliantly in the
orientation image (Fig. 6b) where there are trigonal arrangements of three {100} and three {111} sectors. The tear-shaped
{100} sectors indicate a changing habit from {100} to {111} in
the presence of S2O322. A complete interpretation of these
images requires further separation of the chiroptical properties
that can not be accomplished in a birefringent crystal with
Metripol alone (Section 4.3.1).
4.1.3. 1,8-Dihydroxyanthraquinone. Another example of an
optically anomalous crystal is 1,8-dihydroxyanthraquinone (1,
Fig. 8). The crystals form square plates and are well-refined in
the space group P41(3); they should not be birefringent when
viewed along the optic axis. However, the crystals display
complex patterns of birefringence as shown in Fig. 7a. Despite
the variance in |sin d| across the crystal plate, the extinction is
relatively simple, indicating biaxial sectors related to one
another by 90u rotations. Here again, by itself, Metripol did not
aid in our understanding of the apparent reduction in optical
symmetry. A deeper analysis requires the incorporation of
chiroptical effects (Sections 4.3.1/4.5.1).
Fig. 7 Metripol LB micrographs of 1,8-dihydroxyanthraquinone
(1). (a,b) orthoscopic illumination. (c,d) conoscopic illumination.
(a,c) |sin d| (b,d) w(deg).
Chem. Soc. Rev., 2004, 33, 514–525
519
The new methods of analysis work equally well in conoscopic
as well as orthoscopic illumination.42 The conoscopic image
corresponding to the |sin d| map (Fig. 7c) shows the
birefringence increasing from the positions of the optic axes
(central nodes). Moving outward, |sin d| passes at least four
successive minima and maxima. The orientation image shows
alternating bands with 90u relationships between adjacent
bands. This is a consequence of the fact that the intensity
formula at a fixed wavelength results in an ambiguity in the
relationship between |sin d| and w; each time |sin d| passes
through zero, the calculated w undergoes a 90u phase shift. The
helical nature of the orientation map (Fig. 7d) is a manifestation of the so-called Airy’s spiral in a biaxial crystal.43 Geday
and Glazer showed that in a uniaxial crystal the magnitude of
the optical rotation about the optic axis can be read directly
from the conoscopic orientation image by measuring the
inclination of w ~ h(w0).42 For small rotations (r), it was shown
that the inclination of the zero orientation was proportional to
21/2rL where L is the thickness. Whether this approximation
is valid for biaxial, circularly dichroic samples is a matter for
further investigation.
Fig. 9 LD micrographs of the metastable, orthorhombic polymorph
of 1,8-dihydroxyanthraquinone. (a) tanh e, (b) w’ (deg).
4.2. Linear dichroism
4.2.1. 1,8-Dihydroxyanthraquinone. The tetragonal form of
1,8-dihydroxyanthraquinone (1; Fig. 8) displays anomalous LB
frequently occur in dye single crystals. We have made an
extensive study of the process of dyeing crystals,44 especially
K2SO4.45
The dye amaranth (2) stains the {110} and {010} growth
sectors of K2SO4 (Fig. 10). The tanh e image (Fig. 11a) shows
Fig. 10 Schematic of an amaranth dyed K2SO4 crystal. (a) Crystal as
grown, (b) (010) slice.
Fig. 8 Dyes.
but not anomalous LD. The square plates precipitate simultaneously with an orthorhombic polymorph (space group
Pca21) having hair-like needle habits. These needles must be
linearly dichroic, and indeed, the dichroism and the orientation
of the most strongly absorbing direction can be displayed as
false color maps with Metripol as in Figs. 9a and 9b. This is
achieved by choosing a filter that passes monochromatic light
in the absorption band of the crystal and by removing the
circular analyzer as described above (Section 3.2).
4.2.2. Dyed K2SO4. In the same way that linear birefringence can be produced in crystals of NaClO3 or NaBrO3
through mutual admixing, colorless crystals such as K2SO4 can
be made linearly dichroic by including in the growth solution
dyes that are oriented and overgrown by the transparent host.
In this way, we obtain dyes in single crystal register, uncomplicated, in many cases, by the collective interactions that
520
Chem. Soc. Rev., 2004, 33, 514–525
that the magnitude of the absorption anisotropy is comparable
in the two sectors - though greater in the center of {010}
suggesting that rapid early growth leads to higher anisotropy, a
likely consequence of a selective, kinetically controlled process
– while the anisotropy disappears near the intersection of the
sectors. Since the dye adopts orthogonal orientations in the two
sectors (Fig. 11b), the regions of overlap in a crystal of finite
thickness lead to isotropic absorption.
4.3. Circular birefringence
The first topograph of optical rotation measured along a
birefringent direction in a crystal was of the mineral
langbeinite, K2Cd2(SO4)3.46 This was achieved via a modification of the S-HAUP technique that also involves a birefringence modulation by tilting the crystal about an axis
perpendicular to the wave vector. However, given the timeconsuming measurement process requiring successive scanning
and tilting, the resolution of the image was low. Topographs of
the spontaneous Faraday effect in FeBO3 were produced
shortly thereafter by making measurements below the Curie
vice versa.49 Bijvoet and coworkers50 ultimately established on
the basis of the anomalous dispersion of X-rays that
homochiral crystals of NaClO3 and NaBrO3 do indeed have
opposite signs of optical rotation. Why?
An experiment that might lend some insight into this
problem would exploit the miscibility of the two halates to
determine whether the contributions from the components to
the rotatory power were independent of one another. This
question has not been addressed because the mixed crystals
display the anomalous birefringence described in Section 4.1.2
that precluded measurements of optical rotation. Annealing
near the melting temperature can remove the anomalous
birefringence and restore P213 symmetry. Several groups have
measured the rotation in isotropic, annealed mixed crystals but
with compositions exceeding no more than 15% bromate,51
thus only a small part of the mixed crystal space was analyzed
in crystals in which information encoded during growth was
destroyed. We have studied the development of LB in the
mixed crystals across the entire composition range in some
detail but OR had been out of reach.52
Fig. 11 Metripol LD micrographs of amaranth (2) dyed K2SO4. (a) tanh
e, (b) w’ (deg). Orientation azimuths have been plotted on micrograph.
temperature.47 Both positive and negative optically rotatory
domains were revealed in topographs of triglycine sulfate
[(NH2CH2COOH)3?H2SO4].48
Along non-birefringent crystallographic directions, Metripol
can be used to image optical rotation directly. A reduced
optical path is employed in which only the quarter wave plate is
removed. This is illustrated for the case of a (0001) slice of
quartz. Quartz crystals are usually marked by Dauphine twins
related by a 180u rotation about the hexagonal axis, and Brazil
twins where dextro- and levorotatory domains are reflected
across {11-20}, such as those in Fig. 12.
Fig. 13 S-HAUP generated topographs of optical properties of a
mixed halite (NaCl0.5Br0.5O3, 0.435 mm thick) crystal. (a) transmission
(arbitrary units, a.u.), (b) phase difference d, (c) extinction h(deg),
(d) optical rotation Q(deg).
Fig. 12 Enantiomorphous Brazil twinning in a (0001) plate of quartz.
(a) Optical rotation plotted as Q(deg). (b) Idealized representation of
quartz crystal and the slice from which the micrograph was made.
4.3.1. NaClxBr12xO3. NaClO3 and NaBrO3 have long
puzzled crystallographers. Imagine Marbach’s consternation
when in 1856 he observed that levorotatory crystals of NaClO3
nucleated dextrorotatory crystals of its isomorph NaBrO3 and
In order to image OR along birefringent directions, S-HAUP
is required. Shown in Figs. 13a–d are four S-HAUP topographs displaying the separation of the transmission, retardation, extinction, and optical rotation of a mixed halate crystal.
The as-grown crystal topographs are chaotic. The presence of
dextro- and levorotatory domains may indicate the segregation
of BrO32 and ClO32 on the microscale.
4.3.2. 1,8-Dihydroxyanthraquinone. Tetragonal crystals of
1,8-dihydroxyanthraquinone (1) were examined by the
S-HAUP method. They appeared to have both dextro- and
levorotatory domains (Fig. 14c), however, the poor quality of
Fig. 14 S-HAUP topographs of 1,8-dihydroxyanthraquinone. (a) Phase difference d; (b) Extinction h(deg); (c) Optical rotation Q(deg).
Chem. Soc. Rev., 2004, 33, 514–525
521
Fig. 15 CD micrographs of the tetragonal form of 1,8-dihydroxyanthraquinone. (a) and (b) are different crystals.
the images, resulting from non-uniform extinction across the
crystal plate, made interpretation of the heterochiral domains
speculative. Clearly, another tool would be required to further
explore this contrast (Section 4.4).
faces are unusual in the D2h-symmetric crystals because they
are chiral. As such, the biaryl dye must be adsorbed enantioselectively to these faces. An idealized drawing of such a dyed
crystal is shown in Fig. 16.
4.4. Circular dichroism: 1,8-dihydroxyanthraquinone
CD imaging is well suited for studying the bright orange
crystals of 1,8-dihydroxyanthraquinone (1). From our
S-HAUP micrographs, we suspected that stress from enantiomorphous twinning resulted in the observed anomalous
birefringence. Indeed, CD micrographs in Fig. 15, recorded at
530 nm, show mirror image domains as red (CD is positive) and
blue (CD is negative) heterochiral pinwheels. These images are
independent of the orientation of the microscope stage, which is
the surest way to rule out linear biases in the optical train.
4.5. Anomalous azimuthal rotation and anomalous circular
extinction
4.5.1. Dyed K2SO4. We set out to use the S-HAUP technique for imaging optical rotation (OR) in simple centrosymmetric crystals that had adsorbed, oriented, and overgrown
chiral dye molecules. Chiroptical effects such as OR and CD
were expected when equilibrium racemic mixtures of dyes
selectively recognized chiral facets of achiral crystalline hosts.
Crystals of K2SO4 grown in the presence of trypan blue (3)
were colored in the {110} and {111} growth sectors. The {111}
Fig. 16 Schematic of a trypan blue (3) dyed K2SO4 crystal. (a) Crystal
as grown, (b) (010) slice.
S-HAUP topographs of a K2SO4/3 (010) section are shown
in Fig. 17, where the dyed regions exhibit contrasting signals
consistent with crystal symmetry. The first row in Fig. 17
represents the phase d, which changes sign when the fast and
slow axes are exchanged. In the second row, contrary to
expectation, the sign of the apparent OR Q changes with sample
Fig. 17 Optical topographs of K2SO4/3 (010) plate (y90 mm) using the S-HAUP technique. First row: Phase difference d, l ~ 670 nm for various
crystal orientations. The shape of the sample seen in the images in this row is a guide for the eye as the sample is reoriented in the columns of this
table. Second row: Apparent optical rotation Q. The signs of the quantities refer to the top-left quadrant in the first image. ‘‘Rotation’’ ~ 90u
rotation about wave vector. ‘‘Flip’’ ~ 180u rotation about horizontal or vertical axes of the crystal plate.
522
Chem. Soc. Rev., 2004, 33, 514–525
reorientation, that is, the sign of the effect changes whenever
the sample is turned by 90u about the wave vector (‘‘rotation’’)
or rotated 180u around the vertical or horizontal axes
perpendicular to the wave vector (‘‘flip’’). Intrinsic OR (and
CD) would be invariant to these transformations.
What are the origins of the signals in Fig. 17? Dye molecules
absorb light anisotropically. In an isotropic medium or along
the optic axis of an anisotropic crystal, the absorbance (a) along
the induced dipole of the dye affects the electric field component
along the dipole according to (102a)1/2. The Jones matrix
describing the anisotropic absorption of a dipole in its own
reference system with the absorption strongest along the y-axis
and zero along the x-axis of a Cartesian reference system is then:
1
0
:
~
MDipole
Absorption
0 10{a=2
In an anisotropic host in a birefringent direction, the
absorption can only be measured along the eigenmodes e’
and e@ (Fig. 18). If the dye molecules are inclined to e@ of the
rotatory effect is considered positive when rotation is clockwise
facing the light source (which means that an induced dipole
inclined clockwise from the electric field vector of the incoming
wave corresponds to a negative or levorotation), we find
the following Jones matrix describing AAR as a result of the
absorption by the dye measured along the eigenmodes of the
host crystal:
1 Q
1:15a00
MAAR &
, {Q~
, a00 %1; b~450 ,
1{1:15a00
Q 1
(Q in rad in above equation). This matrix is very similar to that
of OR except that in this case the off diagonal elements have
the same sign.
4.5.2. Dyed LiKSO4. An effect analogous to AAR can be
revealed by the CDIM (now more appropriately called circular
extinction imaging microscope, CEIM) that we call anomalous
circular extinction (ACE). ACE is strong in LiKSO4 crystals
that have oriented and overgrown the dye Chicago sky blue
(4).54 Dyed, hexagonal crystals (P63) are represented in Fig. 20.
Fig. 18 Projection of the absorption of a dipole on the eigenmodes of
a crystal.
host by an angle b, the projections of absorption, a’ and a@, on
the eigenmodes leads to LD. The angle b is obtained from
rffiffiffiffiffi
a0
tan b~ 00
a
The Jones matrix is now recast as follows (assuming small
absorption53 a % 1, (102a)1/2 # (1 2 ln(10)a/2) # (1 2 1.15a),
and rotation matrix R for b # 45u):
1{x {x
Eigenmodes
MAbsorption
~RT MAbsorption R&
,
{x 1{x
x~ ln (10)a=4%1:
The effect of absorption by the dipoles can be described as a
projection of the light wave’s field vector E along and normal to
the direction of the induced dipole (Fig. 19). The absorption
Fig. 19 Model for azimuthal rotation based on absorption of dipoles
inclined towards the eigenmodes of the host. It is assumed that the
perturbation to the refractivity of the host crystal due to the dye
molecules is small.
along the dipole is derived from a’ and a@: adipole ~ a@/cos2b.
Vector addition of the E field components along the dipole
yields a polarization that appears to be rotated by the angle
Q ~ (b’ 2 b). We call this phenomenon anomalous azimuthal
rotation (AAR).
Consistent with experiment, the angle Q changes sign upon
flipping the sample 180u around one of the eigenmode directions and upon rotating the sample 90u about the wave vector
of the incident light. Abiding by the convention that an optical
Fig. 20 Idealized representation of a Chicago sky blue (4) dyed
LiKSO4 crystal. (a) As grown crystal and (b) (100) slice.
The (001) growth sectors were heavily colored whereas the
(011) growth sectors were less optically dense by a factor of 4.
When viewed through the (100) face with the CEIM described
in Section 3.4, the crystals showed a strong differential
transmission near the absorption maximum of the dye in the
lightly dyed sectors. The micrograph revealed four quadrants
with adjacent sectors having opposite sign, a consequence of
the well-known enantiomorphous twinning of LiKSO4 previously revealed by X-ray topography (Fig. 21).55
In the dyed LiKSO4 sample that we now know to possess LB
and ACE, the amplitudes A’¡ are then:
1
0
Q
"
#
"
#
sin x
eix
0
A
x
C p1ffiffiffi 1
B
E0 ~
~
~M
A0+ ~ 1+
A
@Q
AAR,LB
A02+
2 +i
sin x e{ix
x
0
1
0
1
Q
Q
ix
cos xzi sin x+i sin x
e +i sin x
1
x
x
B
C pffiffiffi
B
C.
E0 ~@
@Q
A
A
Q
2
sin x+ie{ix
sin x+i cos ({x)+ sin ({x)
x
x
1
pffiffiffi E0
2
The intensities follow from:
1
. 0
0
A1+ zA02+
A02+ )~
A0+
A+ ~ E02 (A01+
2
Q
E02 fcos2 xz( sin x+ sin x)2 g
x
0
I+
sin2 x
sin2 x
~1+2Q
zQ2 2
2
x
x
E0
Chem. Soc. Rev., 2004, 33, 514–525
523
Fig. 21 Circular extinction micrographs of LiKSO4 dyed with 4 resulting from anomalous circular extinction. (a) Center of crystal, (b) Crystal
rotated by 90u, (c) Crystal flipped around a vertical axis. Here, one can see that the ACE is invariant with respect to rotation but anti-symmetric with
respect to the direction of the wave vector.
and we find indeed a difference in light intensities for the
transmission of left and right circularly polarized light:
IR0 {IL0
sin2 (d=2)
~4Q
I0
d=2
Thus, differential circular extinction contrast can result, but
only if the sample is birefringent and the phase d | n2p, where
n ~ 1, 2, 3….
5. Outlook
In principle, a single instrument could be constructed to
accurately measure and separate the contributions of LB, LD,
OR, and CD through modifications of the optical path and
mechanically modulated linearly and circularly polarized light
input. We aspire to construct such an instrument, the natural
culmination of the studies discussed herein.
There are a variety of other physical properties of crystals
that have been reduced to images via novel microscopies. The
work of Hulliger and coworkers in the development of
scanning pyroelectric microscopy and phase sensitive second
harmonic generation is particularly imaginative.56 While these
techniques do not necessarily fall within the theme that we have
emphasized in this review – the unfolding of convolved optical
properties through the analysis of intensity measurements in
polarized light – these other techniques are complementary in
the insights that they yield and might certainly lead to a much
deeper understanding of the materials that we have discussed
here.
Acknowledgements
This work was supported by the USA National Science
Foundation (BK), the Petroleum Research Fund of the
American Chemical Society (BK, WK), and the University
of Washington Royalty Research Fund (WK). KC received
fellowship support from the American Chemical Society and
the International Centre for Diffraction Data. We are most
grateful for the contributions of the individual researchers
acknowledged in the studies cited, especially Guy Crundwell
for preparing the halite thin sections, Loyd Bastin and Morten
Andreas Geday for growing K2SO4, and Alex Sodt for
preliminary studies of the chiroptical properties of the mixed
halates.
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